EC 720 - Math for Economists Samson Alva Department of Economics, Boston College October 12, 2011

Problem Set 3 Solutions 4. Hotelling’s Lemma (a) The Lagrangian for the firm’s problem is L(y, k, l, λ) = py − rk − wk + λ[f (k, l) − y]. (b) According to the Kuhn-Tucker theorem, the values y ∗ , k ∗ , and l∗ that solve the firm’s problem, together with the associated value λ∗ for the multiplier, must satisfy the first-order conditions L1 (y ∗ , k ∗ , l∗ , λ∗ ) = p − λ∗ = 0, L2 (y ∗ , k ∗ , l∗ , λ∗ ) = −r + λ∗ f1 (k ∗ , l∗ ) = 0, and L3 (y ∗ , k ∗ , l∗ , λ∗ ) = −w + λ∗ f2 (k ∗ , l∗ ) = 0, the constraint L4 (y ∗ , k ∗ , l∗ , λ∗ ) = f (k ∗ , l∗ ) − y ∗ ≥ 0, the nonnegativity condition λ∗ ≥ 0, and the complementary slackness condition λ∗ [f (k ∗ , l∗ ) − y ∗ ] = 0. (c) Assume that the output and input prices p, r, and w and the production function f are such that it is possible to solve uniquely for the values of y ∗ , k ∗ , l∗ , and λ∗ in terms of the parameters p, r, and w. Then the profit function, defined as π(p, r, w) = max py − rk − wl subject to f (k, l) ≥ y, y,k,l

can be evaluated as π(p, r, w) = py ∗ (p, r, w) − rk ∗ (p, r, w) − wl∗ (p, r, w)

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or, using the complementary slackness condition, π(p, r, w) = py ∗ (p, r, w) − rk ∗ (p, r, w) − wl∗ (p, r, w) +λ∗ (p, r, w){f [k ∗ (p, r, w), l∗ (p, r, w)] − y ∗ (p, r, w)}, where y ∗ (p, r, w) is the firm’s supply function, k ∗ (p, r, w) and l∗ (p, r, w) are the factor demand curves, and λ∗ (p, r, w) is the function describing the associated values of the Lagrange multiplier. The envelope theorem says that in differentiating this expression for the profit function through by each argument, one can ignore the dependence of y ∗ , k ∗ , l∗ , and λ∗ on those parameters, and simply write π1 (p, r, w) = y ∗ (p, r, w), π2 (p, r, w) = −k ∗ (p, r, w), and π3 (p, r, w) = −l∗ (p, r, w), which are the same results described by Hotelling’s lemma. 5. Shephard’s Lemma (a) The Lagrangian for this problem is L(k, l, µ) = rk + wl − µ[f (k, l) − y]. (b) According to the Kuhn-Tucker theorem, the values k ∗ and l∗ that solve the firm’s problem, together with the associated value µ∗ for the multiplier, must satisfy the first-order conditions L1 (k ∗ , l∗ , µ∗ ) = r − µ∗ f1 (k ∗ , l∗ ) = 0 and L2 (k ∗ , l∗ , µ∗ ) = w − µ∗ f2 (k ∗ , l∗ ) = 0, the constraint L3 (k ∗ , l∗ , µ∗ ) = f (k ∗ , l∗ ) − y ≥ 0, the nonnegativity condition µ∗ ≥ 0, and the complementary slackness condition µ∗ [f (k ∗ , l∗ ) − y] = 0. (c) Assume that the output requirement y, the input prices r and w, and the production function f are such that it is possible to solve uniquely for the values of k ∗ , l∗ , and µ∗ in terms of the parameters r, w, and y. Then the cost function,

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defined as c(r, w, y) = min rk + wl subject to f (k, l) ≥ y, k,l

can be evaluated as c(r, w, y) = rk ∗ (r, w, y) + wl∗ (r, w, y) or, using the complementary slackness condition, c(r, w, y) = rk ∗ (r, w, y) + wl∗ (r, w, y) − µ∗ (r, w, y){f [k ∗ (r, w, y), l∗ (r, w, y)] − y}, where k ∗ (r, w, y) and l∗ (r, w, y) now represent the conditional factor demand curves and µ∗ (r, w, y) is the function describing the associated values of the Lagrange multiplier. The envelope theorem says that in differentiating this expression for the cost function through by each argument, one can ignore the dependence of k ∗ , l∗ , and µ∗ on those parameters, and simply write c1 (r, w, y) = k ∗ (r, w, y) and c2 (r, w, y) = l∗ (r, w, y), which are the same results described by Shephard’s lemma. 6. Roy’s Identity (a) The Lagrangian for this problem is L( c1 , c2 , λ) = U (c1 , c2 ) + λ(I − p1 c1 − p2 c2 ). (b) According to the Kuhn-Tucker theorem implies that, the values c∗1 and c∗2 that solve the consumer’s problem, together with the associated value λ∗ for the multiplier, must satisfy the first-order conditions L1 (c∗1 , c∗2 , λ∗ ) = U1 (c∗1 , c∗2 ) − λ∗ p1 = 0 and L2 (c∗1 , c∗2 , λ∗ ) = U2 (c∗1 , c∗2 ) − λ∗ p2 = 0, the constraint L3 (c∗1 , c∗2 , λ∗ ) = I − p1 c∗1 − p2 c∗2 ≥ 0, the nonnegativity condition λ∗ ≥ 0, and the complementary slackness condition λ∗ (I − p1 c∗1 − p2 c∗2 ) = 0. (c) Assume that income I, and goods prices p1 and p2 , and the utility function U are 3

such that it is possible to solve uniquely for the values of c∗1 , c∗2 , and λ∗ in terms of the parameters I, p1 , and p2 . Then the indirect utility function, defined as v(p1 , p2 , I) = max U (c1 , c2 ) subject to I ≥ p1 c1 + p2 c2 , c1 ,c2

can be evaluated as v(p1 , p2 , I) = U [c∗1 (p1 , p2 , I), c∗2 (p1 , p2 , I)] or, using the complementary slackness condition, v(p1 , p2 , I) = U [c∗1 (p1 , p2 , I), c∗2 (p1 , p2 , I)]+λ∗ (p1 , p2 , I)[I−p1 c∗1 (p1 , p2 , I)−p1 c∗2 (p1 , p2 , I)], where c∗1 (p1 , p2 , I) and c∗2 (p1 , p2 , I) define the Marshallian demand curves for the two goods and λ∗ (p1 , p2 , I) describes the associated values for the Lagrange multiplier. The envelope theorem says that in differentiating this last expression for the indirect utility function through by each argument, one can ignore the dependence of c∗1 , c∗2 , and λ∗ on those parameters, and simply write v1 (p1 , p2 , I) = −λ∗ (p1 , p2 , I)c∗1 (p1 , p2 , I), v2 (p1 , p2 , I) = −λ∗ (p1 , p2 , I)c∗2 (p1 , p2 , I), and v3 (p1 , p2 , I) = λ∗ (p1 , p2 , I). Dividing the first and second of these equations by the third leads directly to a statement of Roy’s identity. 7. The Slutsky Equation (a) The Lagrangian for this problem is L(c1 , c2 , µ) = p1 c1 + p2 c2 − µ[U (c1 , c2 ) − U¯ ]. (b) According to the Kuhn-Tucker theorem, the values c∗1 and c∗2 that solve this problem, together with the associated value µ∗ for the multiplier, must satisfy the first-order conditions L1 (c∗1 , c∗2 , µ∗ ) = p1 − µ∗ U1 (c∗1 , c∗2 ) = 0 and L2 (c∗1 , c∗2 , µ∗ ) = p2 − µ∗ U2 (c∗1 , c∗2 ) = 0, the constraint L3 (c∗1 , c∗2 , µ∗ ) = U (c∗1 , c∗2 ) − U¯ ≥ 0, the nonnegativity condition µ∗ ≥ 0, 4

and the complementary slackness condition µ∗ [U (c∗1 , c∗2 ) − U¯ ] = 0. (c) Assume that income I, goods prices p1 and p2 , the utility function U , and the utility level U¯ are such that it is possible to solve uniquely for the values of c∗1 , c∗2 , and µ∗ in terms of the parameters U¯ , p1 , and p2 . Now the functions c∗1 = h∗1 (p1 , p2 , U¯ ) and c∗2 = h∗2 (p1 , p2 , U¯ ) describing the optimal choices are the Hicksian demand curves for the two goods. Along with these functions, define the expenditure function e(p1 , p2 , U¯ ) = min p1 c1 + p2 c2 subject to U (c1 , c2 ) ≥ U¯ . c1 ,c2

Under most circumstances, it will be the case that the Marshallian and Hicksian demand curves coincide at the point where I = e(p1 , p2 , U¯ ), so that the income I from the utility maximization problem equals the expenditure required to attain the utility level v(p1 , p2 , I) = U¯ . This result can be summarized by stating that h∗i (p1 , p2 , U¯ ) = c∗i (p1 , p2 , e(p1 , p2 , U¯ )) for all values of p1 , p2 , and U¯ and each i = 1, 2. Differentiating both sides of this expression by pi and using the chain rule on the right-hand side, yields ∂h∗i (p1 , p2 , U¯ ) ∂c∗ (p1 , p2 , e(p1 , p2 , U¯ )) ∂c∗i (p1 , p2 , e(p1 , p2 , U¯ )) ∂e(p1 , p2 , U¯ ) = i + . ∂pi ∂pi ∂I ∂pi However, the envelope theorem, when applied to evaluate the derivatives of the expenditure function, implies that ∂e(p1 , p2 , U¯ ) = h∗i (p1 , p2 , U¯ ) = c∗i (p1 , p2 , e(p1 , p2 , U¯ )). ∂pi Substituting this last expression, together with I = e(p1 , p2 , U¯ ) and v(p1 , p2 , I) = U¯ , into the one above it and rearranging yields the Slutsky equation ∂c∗i (p1 , p2 , I) ∂h∗i (p1 , p2 , v(p1 , p2 , I)) ∂c∗i (p1 , p2 , I) ∗ = − ci (p1 , p2 , I) ∂pi ∂pi ∂I for each i = 1, 2.

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